Reduction in the Resonance Error in Numerical Homogenization II: Correctors and Extrapolation

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• 作者：Antoine Gloria ; Zakaria Habibi
• 关键词：Numerical homogenization ; Resonance error ; Effective coefficients ; Correctors ; Periodic ; Almost periodic ; Random ; 35J15 ; 35B27 ; 65N12 ; 65N15 ; 65B05
• 刊名：Foundations of Computational Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：16
• 期：1
• 页码：217-296
• 全文大小：1,881 KB
• 参考文献：1.A. Abdulle. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul., 4:447–459, 2005.CrossRef MathSciNet MATH
  2.G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23:1482–1518, 1992.CrossRef MathSciNet MATH
  3.G. Allaire and R. Brizzi. A multiscale finite element method for numerical homogenization. Multiscale Model. Simul., 4:790–812, 2005.CrossRef MathSciNet MATH
  4.T. Arbogast. Numerical subgrid upscaling of two-phase flow in porous media. In Numerical treatment of multiphase flows in porous media (Beijing, 1999), volume 552 of Lecture Notes in Phys., pages 35–49. Springer, Berlin, 2000.
  5.I. Babuska and R. Lipton. \(L^2\) -global to local projection: an approach to multiscale analysis. Math. Models Methods Appl. Sci., 21(11):2211–2226, 2011.CrossRef MathSciNet MATH
  6.A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978.
  7.X. Blanc and C. Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Netw. Heterog. Media, 5(1):1–29, 2010.CrossRef MathSciNet MATH
  8.A. Bourgeat, A. Mikelić, and S. Wright. Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math., 456:19–51, 1994.MathSciNet MATH
  9.W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Commun. Comput. Phys., 2:367–450, 2007.
  10.W. E, P.B. Ming, and P.W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., 18:121–156, 2005.
  11.Weinan E. Principles of multiscale modeling. Cambridge University Press, Cambridge, 2011.MATH
  12.Y. Efendiev and T. Y. Hou. Multiscale finite element methods, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009. Theory and applications.
  13.Y.R. Efendiev, T.Y. Hou, and X.H. Wu. Convergence of a nonconforming multiscale finite element method. SIAM J. Num. Anal., 37:888–910, 2000.CrossRef MathSciNet MATH
  14.A.-C. Egloffe, A. Gloria, J.-C. Mourrat, and T. N. Nguyen. Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth. IMA J. Num. Anal., 2014. doi:10.​1093/​imanum/​dru010 .
  15.FreeFEM. http://​www.​freefem.​org/​ .
  16.A. Gloria. An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul., 5(3):996–1043, 2006.CrossRef MathSciNet MATH
  17.A. Gloria. An analytical framework for numerical homogenization - Part II: windowing and oversampling. Multiscale Model. Simul., 7(1):275–293, 2008.CrossRef MathSciNet
  18.A. Gloria. Reduction of the resonance error - Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci., 21(8):1601–1630, 2011.CrossRef MathSciNet MATH
  19.A. Gloria. Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. M2AN Math. Model. Numer. Anal., 46(1):1–38, 2012.CrossRef MathSciNet MATH
  20.A. Gloria. Numerical homogenization: survey, new results, and perspectives. Esaim. Proc., 37, 2012. Mathematical and numerical approaches for multiscale problem.
  21.A. Gloria and J.-C. Mourrat. Spectral measure and approximation of homogenized coefficients. Probab. Theory. Relat. Fields, 154(1), 2012.
  22.A. Gloria, S. Neukamm, and F. Otto. An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. M2AN Math. Model. Numer. Anal., 2014. Special issue 2014: Multiscale problems and techniques.
  23.A. Gloria, S. Neukamm, and F. Otto. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math., 2014. DOI 10.​1007/​s00222-014-0518-z .
  24.A. Gloria and F. Otto. Quantitative results on the corrector equation in stochastic homogenization. arXiv:​1409.​0801 .
  25.P. Henning and D. Peterseim. Oversampling for the Multiscale Finite Element Method. Multiscale Model. Simul., 11(4):1149–1175, 2013.CrossRef MathSciNet MATH
  26.T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134:169–189, 1997.CrossRef MathSciNet MATH
  27.T.Y. Hou, X.H. Wu, and Z.Q. Cai. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput., 68:913–943, 1999.CrossRef MathSciNet MATH
  28.T.Y. Hou, X.H. Wu, and Y. Zhang. Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Comm. in Math. Sci., 2(2):185–205, 2004.CrossRef MathSciNet MATH
  29.V.V. Jikov, S.M. Kozlov, and O.A. Oleinik. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994.CrossRef
  30.S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.), 107(149)(2):199–217, 317, 1978.
  31.H. Owhadi and L. Zhang. Metric-based upscaling. Comm. Pure Appl. Math., 60(5):675–723, 2007.CrossRef MathSciNet MATH
  32.H. Owhadi and L. Zhang. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. Multiscale Model. Simul., 9(4):1373–1398, 2011.CrossRef MathSciNet MATH
  33.H. Owhadi, L. Zhang, and L. Berlyand. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Mathematical Modelling and Numerical Analysis, 2014.
  34.G.C. Papanicolaou and S.R.S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835–873. North-Holland, Amsterdam, 1981.
  35.M. Vogelius. A homogenization result for planar, polygonal networks. RAIRO Modél. Math. Anal. Numér., 25(4):483–514, 1991.MathSciNet MATH
  
• 作者单位：Antoine Gloria (1) (2)
  Zakaria Habibi (2)
  
  1. Université Libre de Bruxelles (ULB), Brussels, Belgium
  2. Project-Team MEPHYSTO, Inria Lille - Nord Europe, Villeneuve d’Ascq, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Computer Science, general
  Math Applications in Computer Science
  Linear and Multilinear Algebras and Matrix Theory
  Applications of Mathematics
  
• 出版者：Springer New York
• ISSN：1615-3383

文摘

This paper is the follow-up of Gloria (Math Models Methods Appl Sci 21(8):1601–1630, 2011). One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio \(\frac{\varepsilon }{\rho }\), where \(\rho \) is a typical macroscopic lengthscale and \(\varepsilon \) is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost-periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g., Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients, and give quantitative estimates in the case of periodic coefficients. Keywords Numerical homogenization Resonance error Effective coefficients Correctors Periodic Almost periodic Random